vo
= “fluid threshold” and ”impact threshold” velocity
V,
=
vs,
=
w
=
A Pf PP
Pf w
= =
= =
=
criteria of Bagnold, ft.,/sec. saltation velocity under load conditions; minimum superficial velocity required to convey horizontally a t finite rate: Il’, without layer of solids forming in bottom of tube, ft./’sec. saltation velocity for single particle; minimum superficial velocity required to convey single particle horizontally without its settling out and lying stagnant in bottom of tube, ft.jsec. solids mass flow rate, lb.,’sec. X sq. ft. of pipe cross section [31’,*V4g~h~ P,) 11’3, feet fluid density, lb./cu. it. particle density, Ib./’cu. ft. fluid viscosity, Ib./ft. X sec. [4gpf(p,, - p f ) : ’ 3 p f z ] 1 ’ 3 , ft./sec.
literature Cited
(1) Ambrose, H. H., Ph.D. thesis, University of Iowa, Iowa City, 1952. (2) Bagnold, R. .4.,“The Physics of Blown Sand and Desert Dunes,” Methuen, London, 1941. (3) Cairns, R. C., Lawther, K R., Turner, K. S., Brit. Chem. Eng. 5, 849-56 (1960) (4) Culgan, J. M., M.S. thesis, Georgia Institute of Technology, Atlanta, May 1952. (5) Durand. R., Compte Rendu des Deuxihmes JourneCs de I’Hydraulique, June 1952.
(6). Durand, R., Proceedings of Minnesota Hydraulics Convention, 1953. (7) Gruca, \V., Combustion 30, 57-61 (November 1958). (8) Hjulstrom, F., Bull. Geol. Inst. C$sala 25 (1936). (9) Hughmark, G. A,, Ind. Eng. Chem. 53, 389-90 (1961). (10) Jordan, D. LV., Brit. J . Appi. Phys. 5, Suppl. 3, S194-8 (1954). (11) Patterson, R. C., Combustion 30, 47-57 (July 1958). (12) Patterson, R. C., Trans. Am. Sot. .Mech. Engrs. 81,43 (January 1959). (13) Schlichting, H . , “Boundary Layer Theory,” Chap. XX, McGraw-Hill, New York, 1955. (14) Sinclair, C. G., “Third Congress of European Federation of Chemical Engineering.“ pp. A68-A76, London, June 1962. (15) Spells, K. E., Trans. Inst. Chem. Ent. (London) 33, No. 2, 79-84 (1955). (16) Thomas, D. G., A.I.Ch.E. J . 8, No. 3, 373-8 (1962). (17) Zenz, F. A , , D.Ch.E,. thesis, Polytechnic Institute of Brooklvn. 1960. (18j-Zenz, F. A,, Ind. Eng. Chem. 41, 2801-6 (1949). (19) Zenz, F. A , , Othmer, D. F., “Fluidization and Fluid-Particle Systems,” Reinhold, New York, 1960. (20j Ibid.,’p. 328. RECEIVED for review November 6, 1962 ACCEPTED April 29, 1963 Presented in part at the Division of Industrial and Engineering Chemistry, 144th Meeting, ACS, Los Angeles, Calif., April 1963. Based on a dissertation submitted to the faculty of the Polytechnic Institute of Brooklyn in October 1960 in partial fulfillment of the requirements for the degree of doctor of chemical engineering.
EFFECT OF ELECTRIFICATION ON T H E DYNAMICS OF A PARTICULATE SYSTEM S
.
L
.
S 0 0 , C‘niversity of Illinois, Cibana, Ill.
The significance of electrification on the dynamics of a gas-solid system was studied. At low temperatures, electrification of solid particles occurs because of impact with a wall. Basic considerations include motion of charged particles with spherical symmetry and axi-symmetry. It i s shown that the pinch effect due to motion of particles charged to the same sign i s insignificant. However, even a very slight charge on the solid particles will have a pronounced effect on concentration distribution in the flow of a gas-solid system. HE dynamics of a particulate system in a gas has been a T s u b j e c c of many intensive studies because of its applications in reactor technology, utilization of metallized propellant in rockets, and many chemical processes. Studies, in general, concern the motion of the solid particles in a turbulent fluid field (7. 76. and their bibliographies) and gas dynamics, including viscous interactions ( 7 7 ) . Although electric charge on the solid particles can be excluded by definition in theoretical analysis or \\.hen dealing truly with a boundless system, electrification of the solid particles alLvays occurs when contact and separation are made bet\veen the solid particles and a \\.all of different materials or similar materials but different surface conditions ( 9 ) . T h e effect of the electric charges on the solid particles \\.as noticed in our measurement of diffusivity of Orlon and poly(viny1 chloride) particles in air by the method of optical recording u i t h successive exposure ( 7 6 ) and optical autocorrelation (77) in 1958; the results are shown in Figure 1. T h e electric charges on the solid particles cause deposition of the solid particles on a wall in a more significant manner than
the gravity effect and are expected to affect the motion of a metallized propellant and its product of reaction through a rocket nozzle and the jet a t the exit of the nozzle. T h e charged solid particles in the jet of a hot gas alro affect radio communications. T h e generality of previous experimental results can be shown to be limited to the extent that the latter are applicable to situations of similar electrification (the nature of similarity is discussed below). Data evaluation should also account for such effects. The following presents aspects of charge distribution and aspects of the dynamics of a gas-solid suspension as affected by the electrification of the solid particles While an ionized gas is always electrically neutral locally (over distances greater than the Debye length), the solid particles suspended in the turbulent field of a nonionized gas may carry charges of one sign. T h e electric charges on the solid particles may be developed by friction or impact Lvith another surface. I n spite of our inabilitv to predict accurately the charge transfer, which is only knoivn to be related to the VOL. 3
NO. 1
FEBRUARY 1964
75
Fermi energy of the surfaces. the order of magnitude for a maximum electrification may amount to about 1 out of 106 surface atoms (9). For a smooth spherical aluminum particle of 1-micron diameter with 2 X 10'5 atoms per sq. cm. on the surface, it may be charged negatively to 10-'6 coulomb or 10-6 e.s.u. by rubbing it over a n insulator, giving it a charge to mass ratio of 0.07 coulomb per kg. This corresponds to an electrical intensity of 3.6 X 106 volts per meter, \vhich is below the breakdown voltage in the air. Even a charge to mass ratio less than this amount has tremendous influence on the dynamics of a particulate system. A group of particles carrying net charges of similar signs in an uncharged gas is not an equilibrium system. This is in addition to the general nature of nonequilibrium in the gassolid suspension ( 7 7 ) .
where e is the permittivity, q , is the charge a t point n, r, and r t are distances from the chosen origin of the system, and 1, is a unit vector connecting point charge at r, to a point at r t . For a large number of particles, or where continuum approximation is applicable (72). (3)
where p e is the charge density, and for a uniform size of particles and the charge per particle y,
Dynamics of a System of Charged Solid Particles
where n p is the number density of solid particles. p p is the mass
Contributions of charges on the solid particles (in a neutral gas or vacuum) to the dynamics of a gas-solid suspension will include elementary relations of electrostatics and electrodynamics ( 4 ) . T h e forces and moments acting on a solid particle consist of those due to the net charge, electric dipole (permanent or induced dipoles depending on the material) in the electric field due to the charged particles and external field, and magnetic dipole in the induced magnetic field. Neglecting effects of magnetic dipoles. the force, F, acting on a solid particle is given by:
of particles per unit volume, and
F = q(E
+ v X B) + v ( ~ , E )
(:)
ratio of each particle. 'The above volume integral also applies to the electric field in a dielectric. Because of the charge on a wall or boundary. the additional electric field is given by: (5)
where be is the surface charge density and A is the area of the surface, T h e current density due to the mass motion of the charged particle is :
(1)
where q is the charge, E is the electric intensity, v is the velocity, and p is the dipole moment. T h e contribution d u e to electric dipoles is small in the cases under consideration. T h e electric field at point t produced by distributed particles is given by :
T h e relation of the fields is given by the well-known electromagnetic equations. Inclusion of the above forces and work done by these forces in the energy equation of a two-phase system (72) gives all the
0
50pGlass
0 I00 Orion 0200 ,' V'OO,,
.05
wc
VI53000
VI29000 J
'
MT
O
'
W
D. Re.
dp.
Stream diffuslvity d p x u'fu Diameter of particles of turbulence
47, 76
I&EC FUNDAMENTALS
V83500
Numerical ore Reynolds numbers of square pipe 3"x 3:' top E, bottom aluminurn,
Particle diffusivity as measured b y Kadambi, 1958
Figure 1 .
is the charge to mass
p. u.
I], pP.
Density Kinematic viscosity Scale of turbulence of stream Density of solid material
basic relations for a gas-solid suspension. This is sufficient for the case in which the gas is nonionized and the solid particles are charged. Further details will be needed if the gas phase is ionized (74). T h e electric field as given by Equation 2 represents a new complication beyond the usual method of analysis of magnetohydrodynamics. 1,eaving the more general solution for future studies, we now conbider a feu simple basic situations to develop some understanding of the ph\sical phenomena.
-4 p p / p P . T h e quantity p , / P , is none other than [ l - (fraction void)] of the particulate system. Therefore, for the par ticd a t e system of our interest, the dipole effect due 10 self-field is negligible. In this case. integration after nondimensionalizing gives, for a particle a t radius R initially:
where
I*
=
t* =
Simple Symmetric Systems
Let us consider a large number of uniformly charged solid particles emerging from a spherically symmetrical density distribution n i t h the external constraint, which initially kept the particles rvithin a radius R,. suddenly removed. \$’e ask for the spreading of the solid particles and its physical effects. For the sake of simplicity, w e further assume that the particles ar? released in a vacuum [although viscous drag in a gas can be easily accounted for ( I - / ) ] Csing the Lagrangian frame of reference, the momentum balance gives? in the r-direction, m
dv - = qE dt
+ adr ( $ E )
(7)
-
where
r/R.
Further integration gives? for t* = t = 1 a t t * = 0:
.\/ATf&l&)i/ZzR3, and r*
+ In [.\/F +.\/Ti]
drFp-:fj
(16)
Equations 1 5 and 16 give the position and velocity at given time t* of each particle initially a t radius R. Each particle ) 2 / 2 i i R a t infinity reaches a n asymptotic velocity of d\/-Tfi(,j/& (Figure 2). T h e density distribution can be visualized as follows: If the initial density distribution is such that \ve have a uniform sphere, the above relations show that the cloud expands uniformly, giving a uniform distribution all the time and, finally, the radius of the system expands at a constant speed. If the initial distribution is uniform in a spherical shell. the re>ult of its expansion is a uniform hollow sphere with the inside radius preserved and outer radius as given by Equation 16. Because the collision among the particles does not occur in this aysteni, final denaity distribution can be obtained from the initial density distribution by superposition. Because of the spherical symmetry the AMax\vell electromagnetic equations give ( d ) , for magnetic induction B = 0>
for
and
where T~ is the volume of a solid particle, P is the polarization, p p is the density of the solid material, k is the dielectric constant: and velocity c is in the r direction. Equation 7 together with initial conditions gives the complete solution. T h e simplicity of the chosen system makes the solution posbible by expressing v = dr/dt
which is none other than the continuity equation of the cloud of solid particles; D is the electric displacement. The electromagnetic energy ( I O ) of this system is just the electrostatic energy. An energy balance of the system and the surroundings shows that the decrease in the electromagnetic energy as the whole system expands gives rise to an increase in kinrtic energy of the particles. T h e system therefore suffers no loss
(11)
Substitution into Equation 7 gives, for a particle initially a t radius R,
where M E = M , ( R ) at t = 0. where u = 0. It is seen that the dipole effect is due to the gradient of electric intensit) which, in this case, produces a deceleration. T h e ratio of the force due to dipole Fd to that due to electrostatic repulsion. F,, is
which for uniform distribution lvithin r = R is
which gives for very low dielectric constant ( k
2 l ) , F,i F ,
-
( i ) ( k - 1 ) - and for very high dielectric constant Fd F ,
Figure 2. Position and velocity of spherically symmetric motion of electrified solid particles VOL.
3
NO.
1
FEBRUARY
1964
77
via electromagnetic waves. The Poynting flux of the electromagnetic radiation given by E X H is zero. This case involving spherical symmetry is the only situation in which there is no electromagnetic radiation, Next. we take an axi-symmetric system with charged particles moving axially (in the z direction) a t velocity L’, initially constrained within radius R,, and then the radial constraint is removed a t t = 0. The field a t radius I is equal to that due to a line source along the axis of linear charge density, A, where
=
(;)
.M,’(R)
(;)
=
.ME’
where M,’ is the mass of particles per unit axial length within the radius R. T h e field a t radius R gives the acceleration due to electrostatic repulsion from the core (74):
The ratio of the force due to dipole to that due to electrostatic repulsion is now [ 3 ( k - l ) / ( k 2 ) ] ( . W , ” r i ~ ~ * p ? , ) ( p ? , ~and ~) the dipole effect due to self-field is again negligible for cases of our interest. Equation 19 shows that in the absence of turbulence and other field forces. a charged gas-solid suspension in a pipe will eventually settle a t the wall bv the electrostatic force. Because of axial velocity, a magnetic field is induced in the @-direction ( 7 9 ) . T h e radial force on a particle amounts to:
+
Equation 25 gives the change in the total kinetic energy per unit length
The electromagnetic energy per unit length
inside of R is
for a uniform density of the solid particles to begin with. since p?, is always uniform, for the same reason as in the spherical system. Thus, the total electromagnetic energy is constant inside the radius Ro‘. T h e acceleration of the particles is. therefore. due to the decrease of electromagnetic energy outside of the system. I n this case the Poynting flux, s, in energy per unit area per unit time points to the dowm stream of the cvlindrical system.
T h e total axial flux of energy is now:
in energy per unit time
Turbulent Flow of Charged Gas-Solid Suspension
For constant p p , and inside ofradius r. we have: Be
= p
(21 1
Jr/2
where p is the permeability of the particle field. T h e equation of motion of the particles a t radius 7 in the radial direction a t a steady state, is given by pp
u,
5 d7
= -J
Bg
+ n p qET =
- p
J’
4- pi?
(i)’ & (22)
that is, the particles driven by electrostatic field are sloived down by the induced magnetic field depending on the mass flow. p?, r L , T h e J z term is similar to the magnetic pinch in a plasma. Integration of Equation 22 shoivs that, for J given by Equation 6, the electrostatic repulsion is eliminated by the induced magnetic field only when
Since 1 ‘pc is the square of the speed of light, we see that the electrostatic repulsion is canceled only when the axial velocity is that of light in the medium under consideration. Therefore. the self-pinch effect on a charged gas-solid suspension is usually negligible. To calculate the behavior of this system as the outer boundary of particles expands lvithout constraint. Equations 22 and 18 give :
One situation of interest consists of a gas-solid suspension flowing through a pipe. If the pipe is a conductor or a uniformly’ charged insulator. the Gauss law says that there is no electric field inside of the pipe due to the wall. If the particles are uniformly charged and axi-symmetrically distributed over the pipe. the particle will eventually settle a t the \Val1 if there is no fluid turbulence. Equation 19 sho\vs that for small glass beads in atmospheric air a t a loading of 1 kg. of solid per kg. of air. the acceleration amounts to. a t radius 1 cm. from the axis. 10 times that due to gravity. even for a charge to mass ratio of 0.002 coulomb per kg. The radial component of turbulent intensity of particles. according to the approximation of So0 ( 7 5 ) . \vi11 be to the order of 10 sq. meters per sec.2 for 100-micron particles. This effect can overshadow completely that due to gravity on a gas-solid suspension in a horizontal pipe. Density distribution as determined by So0 (73) may attribute its drift velocity mainly to electric charges on the particles also. In dealing Xvith a turbulent suspension of charged solid particles in a gas flowing in a cylindrical pipe of a conductor. the inside wall of the pipe forms a closed equi-potential surface, regardless of its charge. The particles will be driven toward the wall by mutual repulsion ; a sustained gas-3olid suspension \vi11 have to be due to turbulent diffusion from the \vall. Effect of nonuniform charges on the w d l may shoiv u p in the case of a pipe of insulating materials. however. In dealing \vith the fully developed turbulent pipe floiv of a suspension. the concentration distribution of solid particles is given by :
kvhich integrates to. as a particle changes its radical position. For charged particles. Equation 22 gives 78
l&EC FUNDAMENTALS
Pr
The last term due to induced magnetic field is negligible. If we take the distribution as determined by So0 (73) \
